106 Physical Properties [en. vi 



The total impulse due to A is therefore m a (u a u a '), so that 



2m a (u a -u a ') ........................ (258), 



where the summation extends over all molecules which collide with dS during 

 the time dt. To calculate the value of this sum, we require to know the 

 law of distribution of velocities. 



120. In practice, what is observed is not the pressure at a particular 

 instant, but the pressure averaged over a certain interval of time. In fact 

 p, regarded as a function of the time, would be an infinitely rapidly varying 

 quantity similar to the H described in 70. The quantity which experi- 

 ment enables us to determine is the value of p averaged over a length of 

 time sufficient for these rapid variations to be of no importance. In this 

 way an experimental value of p is obtained which is independent of the time. 



Now the value of p depends on the law of distribution of velocities /, and 

 therefore on H. If we evaluate H as a function of the time we shall find it 

 to be an infinitely- rapidly varying function, but if we average its value over 

 a sufficient interval of time, it has, as was shewn in Chapter III., a constant 

 value ; and this value is its minimum value, which corresponds to the normal 

 state. Hence we see that the value of p which is to be expected from ex- 

 periment may legitimately be calculated on the supposition that the gas is in 

 its normal state throughout. 



INFINITELY SMALL MOLECULES. 



121. The number of molecules which have velocities lying between 

 u and u + du etc., and which will impinge on dS during the interval dt is, as 

 in 27, equal to the number of molecules having velocities satisfying these 

 conditions and lying within an element udSdt of volume at the beginning of 

 the interval dt. 



When the molecules are infinitely small, this number may be taken to be 

 vf(u, v, w) dudvdwudSdt ..................... (259). 



Hence we have, in equation (258), 



'2 l m a u a = dSdtmv \\\f(u, v, w)u?dudvdw ......... (260), 



where the integration extends over all values of u, v and w for which the 

 molecules can collide with dS. In other words, the integration is to extend 

 over all possible values of v and w, and over all positive values of u. 



In a similar way, we find that S mu' in equation (258) is equal to the 



A 

 quantity on the right-hand of equation (260), except that the integration must 



now be understood to extend over all values of u, v and w which are possible 

 for molecules which are just rebounding from collision with the element d$. 



